ECE 486 Control Systems
Lecture 14

Its magnitude is

It is a linear function of $\log \omega$; hence a line of slope $n$ passing through the value $\log |K_0|$ at $\omega = 1$. This line is called a low-frequency asymptote.

In our example above, we had $K_0(j\omega)^{-1}$. Its magnitude plot is shown as in Figure 1.

For Type 1 factor, its phase is

The phase is therefore a constant independent of $\omega$.

In our example above, we had $K_0(j\omega)^{-1}$. Its phase plot is shown as in Figure 2. In this example, the phase is $-90^\circ$ for all $\omega$.

This is the case of a stable real zero. To study $|j\omega\tau+1|$ and $\angle(j\omega\tau+1)$ as a function of $\omega$, we will look at the Nyquist plot as shown in Figure 3.

• For $\omega \tau \ll 1$, $j\omega\tau + 1 \approx 1$.
• For $\omega\tau \gg 1$, $j\omega\tau + 1 \approx j\omega\tau$. In this case, Type 2 factor behaves like Type 1 with $K_0 = \tau, n = 1$.
• The transition from one extreme to the other gives $\omega\tau=1\, \Longleftrightarrow \, \omega = 1/\tau$. This frequency is the breakpoint frequency.

As for the magnitude plot,

• For small $\omega$ below breakpoint, $M \approx 1$, a horizontal line.

• For large $\omega$ above breakpoint,

  $$\begin{align*} \log M &\approx \log |j\omega\tau| \\ &= \log \omega\tau \\ &= \log \tau + \log \omega. \end{align*}$$

  This gives a line of slope $1$ passing through the point $(1/\tau,1)$ on a plot of $\log$ - $\log$ scale. Note these are just asymptotes; the actual value of $M$ at $\omega = 1/\tau$ is $\sqrt{2}$.

Figure 4 shows for a stable real zero, the magnitude slope “steps up” by $1$ at the breakpoint.

As for the phase plot, as shown in Figure 3,

• For small $\omega$ (below breakpoint), $\phi \approx 0^\circ$.

• For large $\omega$ (above breakpoint),

  $$\begin{align*} \phi &\approx \angle(j\omega\tau) \\ &= 90^\circ. \end{align*}$$

• At breakpoint ($\omega\tau=1$),

  $$\begin{align*} \phi &= \angle(j+1) \\ &= 45^\circ. \end{align*}$$

  

  Figure 5: Phase plot of $j\omega\tau + 1$

Figure 5 shows for a stable real zero, the phase “steps up” by $90^\circ$ as we go past the breakpoint.

Since this type of factor is the inverse of the one we just discussed, this is a case of a stable real pole.

Its magnitude

and its phase

Therefore the magnitude and phase plots for a stable real pole are the reflections of the corresponding plots for the stable real zero with respect to the horizontal axis,

• at breakpoint frequency, its magnitude plot “step down” by $1$ in magnitude slope and
• its phase plot “step down” by $90^\circ$.

The factor of Type 3 gives stable complex zeros. As our first step, rewrite it in Cartesian form,

Figure 8 below is the Nyquist plot for $0 < \omega < \infty$.

A quick observation here is when $\omega = 0$, The Nyquist plot starts at $1 + 0j$; when $\omega = \omega_n$, Nyquist plot passes through $0 + 2\zeta j$.

When $\omega \to \infty$, we have

• the real part in Equation $\eqref{d14_eq1}$, $\left(1- \left(\frac{\omega}{\omega_n}\right)^2\right) \approx - (\omega/\omega_n)^2 \to - \infty$, quadratic in $\omega$;
• the imaginary part $2\zeta\frac{\omega}{\omega_n} = (2\zeta \frac{1}{\omega_n})\omega \to \infty$, linear in $\omega$.

Now we can begin plotting its magnitude plot.

• for small frequencies $\omega \ll \omega_n$, $M \approx 1 \implies \log M = 0$, resulting in a horizontal line;
• for large frequencies $\omega \gg \omega_n$, $M \approx \left(\frac{\omega}{\omega_n}\right)^2 \implies \log M \approx 2 \log \omega - 2 \log \omega_n$. In this case, the asymptote is a line of slope $2$ passing through the point $(\omega, M) = (\omega_n, 1)$.

Therefore, for factor of Type 3 of a stable complex zero, the magnitude slope “steps up” by $2$ as we go through the breakpoint.

The discussion of its phase plot will be combined with the case of its inverse below.

Similar to what we did to the discussion of Type 2 factors, this case is the inverse of the Type 3 factor we just investigated. It is a case of stable complex poles.

Its magnitude

Its phase

Recall from last time, its magnitude plot (in $M(ω)$-$\frac{\omega}{\omega_n}$ plane) depends on the value of $\zeta$.

The magnitude hits its peak value (for $\zeta < 1/\sqrt{2} \approx 0.707$) occurs when frequency hits resonant frequency $\omega = \omega_r$, where

$$\omega_r = \omega_n \sqrt{1-2\zeta^2} < \omega_n.$$

As for its magnitude plot of Bode, we notice for small $\zeta < \frac{1}{\sqrt{2}}$, the magnitude of

$$\frac{1}{\left(\frac{j\omega}{\omega_n}\right)^2 + 2\zeta \frac{j\omega}{\omega_n} + 1}$$

has a resonant peak at the resonant frequency

$$\omega_r = \omega_n \sqrt{1-2\zeta^2}.$$

Likewise, the magnitude of

$$\left(\frac{j\omega}{\omega_n}\right)^2 + 2\zeta \frac{j\omega}{\omega_n} + 1$$

has a resonant dip at $\omega_r$.

Therefore, for factor of Type 3 of a stable complex pole, the magnitude slope “steps down” by $2$ at the breakpoint.

As for the phase plots of Bode for both Type 3 factors $\left[ \left(\frac{j\omega}{\omega_n}\right)^2 + 2\zeta \frac{j\omega}{\omega_n} + 1\right]^{\pm 1}$, using Equation $\eqref{d14_eq1}$ and Figure 8 above

• for $\omega \ll \omega_n$, $\phi \approx 0^\circ$ (close to positive $1$ on the real axis);
• for $\omega = \omega_n$, $\phi = 90^\circ$ (${\rm Re} = 0,\, {\rm Im} > 0$);
• for $\omega \gg \omega_n$, $\phi \approx 180^\circ$. (Why? Hint: ${\rm Re} \sim -\omega^2$, ${\rm Im} \sim \omega$, take tangent value of this angle, it is almost $0$.)

Therefore, for factor of Type 3 of a stable complex zero, the phase “steps up” by $180^\circ$ as we go through the breakpoint; as $\zeta \to 0$, the transition through the breakpoint gets sharper, almost step-like.

In the case of a factor of Type 3 of a stable complex pole, the phase is multiplied by $-1$, hence the phase “steps down” by $180^\circ$ as we go through the breakpoint.

So far, we’ve only looked at transfer functions with Left Half Plane poles and zeros except perhaps at the origin for Type 1. What about factors with RHP poles and zeros?

Consider two transfer functions below,

We notice

• $G_1$ has a stable pole and zero; $G_2$ has a stable pole but a RHP zero.

• Nevertheless, the magnitude plots of $G_1$ and $G_2$ are the same since

  $$\begin{align*} |G_1(j\omega)| &= \left| \frac{j\omega+1}{j\omega+5}\right| = \sqrt{\frac{\omega^2+1}{\omega^2+5}}, \\ |G_2(j\omega)| &= \left| \frac{j\omega-1}{j\omega+5}\right| = \sqrt{\frac{\omega^2+1}{\omega^2+5}}. \end{align*}$$
• All the difference is in the phase plots.

Indeed, to sketch the phase plot for $G_1$, write it in Bode form

Then apply the rules we have discussed for factors with stable poles and zeros,

• Type 1 factor $\dfrac{1}{5}(j\omega)^0$ has $n=0$, so phase starts at $0^\circ$.
• Type 2 factor of a stable zero has breakpoint at $\omega_n=1$; it makes phase go up by $90^\circ$.
• Type 2 factor of a stable pole has breakpoint at $\omega_n = 5$; it makes phase go down by $90^\circ$.

The phase plot of $G_1(s)$ is therefore

We cannot apply the rules about stable poles and zeros to sketching phase plot for $G_2$. So we need to rework out the study of Nyquist plot of a RHP zero in this case.

The Nyquist plot for $j\omega-1$ is shown in Figure 17.

The new type of behavior of RHP zeros is

• when $\omega \approx 0$, $\phi \approx 180^\circ$ (close to negative $1$ on the real axis);
• when $\omega \gg 1$, $\phi \approx 90^\circ$ (${\rm Re} =-1$, ${\rm Im} = \omega \gg 1$);
• when $\omega \approx 1$, $\phi \approx 135^\circ$.

Therefore for a RHP zero, the phase starts out at $180^\circ$ and goes down by $90^\circ$ through the breakpoint. (Phase is $135^\circ$ at breakpoint.)

For a RHP zero, the phase plot is similar to what we had for a LHP pole — phase goes down by $90^\circ$. However, it starts at $180^\circ$, not at $0^\circ$.

Figure 18 shows the phase plot of $G_2(s)$.

Based off of the discussion from the previous section, among all transfer functions with the same magnitude plot, the one with only LHP zeros has the minimal net phase change as $\omega$ goes from $0$ to $\infty$, hence the term minimum-phase for LHP zeros.